Two equal masses `m_1` and `m_2` moving along the same straight line with velocites `+3 m//s` and `- 5 m//s` respectively collide elastically. Their velocities after the collision will be respectively.

Options

A. `+4 m/s` for both
B. `-3 m/s` and` +5 m/s`
C. `-4 m/s `and` +4 m/s`
D. `-5 m/s` and` +3 m/s`

Answer

To solve the problem of two equal masses colliding elastically, we need to apply the principles of conservation of momentum and the properties of elastic collisions. Let's break down the steps:

### Step 1: Define the variables
Let:
- Mass of both bodies be \( m \) (since they are equal).
- Initial velocity of mass \( m_1 \) (moving to the right) be \( u_1 = +3 \, \text{m/s} \).
- Initial velocity of mass \( m_2 \) (moving to the left) be \( u_2 = -5 \, \text{m/s} \).

### Step 2: Apply the conservation of momentum
The total momentum before the collision must equal the total momentum after the collision. This can be expressed mathematically as:
\[ m u_1 + m u_2 = m v_1 + m v_2 \]
Since the masses are equal, we can divide through by \( m \):
\[ u_1 + u_2 = v_1 + v_2 \]
Substituting the values:
\[ 3 + (-5) = v_1 + v_2 \]
This simplifies to:
\[ -2 = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 3: Use the coefficient of restitution
For an elastic collision, the coefficient of restitution \( e \) is equal to 1. The formula for the coefficient of restitution is:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting the known values:
\[ 1 = \frac{v_2 - v_1}{3 - (-5)} \]
This simplifies to:
\[ 1 = \frac{v_2 - v_1}{8} \]
Thus, we can express this as:
\[ v_2 - v_1 = 8 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( v_1 + v_2 = -2 \) (Equation 1)
2. \( v_2 - v_1 = 8 \) (Equation 2)

We can solve these equations simultaneously. From Equation 2, we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = v_1 + 8 \]
Substituting this into Equation 1:
\[ v_1 + (v_1 + 8) = -2 \]
This simplifies to:
\[ 2v_1 + 8 = -2 \]
Subtracting 8 from both sides:
\[ 2v_1 = -10 \]
Dividing by 2:
\[ v_1 = -5 \, \text{m/s} \]

### Step 5: Find \( v_2 \)
Now, substituting \( v_1 \) back into the expression for \( v_2 \):
\[ v_2 = -5 + 8 = 3 \, \text{m/s} \]

### Conclusion
The velocities after the collision are:
- \( v_1 = -5 \, \text{m/s} \) (moving left)
- \( v_2 = 3 \, \text{m/s} \) (moving right)

Thus, the final answer is:
\[ v_1 = -5 \, \text{m/s}, \quad v_2 = 3 \, \text{m/s} \]